Multilinear maximal operators associated to simplices

We establish $L^{p_1}\times\cdots\times L^{p_k}\to L^r$ and $\ell^{p_1}\times\cdots\times \ell^{p_k}\to \ell^r$ type bounds for multilinear maximal operators associated to averages over isometric copies of a given non-degenerate $k$-simplex in both the continuous discrete settings. These provide natural extensions $L^p\to L^p$ $\ell^p\to \ell^p$ Stein's spherical operator operator, with each these results serving as key ingredient respective proofs.